By the above proporition we may express \(\hat{\Delta }\)  as a \(|W|\times |W|\)  matrix whose entries are rational functions on \(\mathbb {T}^d \subset \mathbb {C}^d\)  expressed via Laurant polynomials.
Classical perturbation analysis shows that \(\lambda \)  is an eigenvalue iff \(det(\hat{\Delta }-\lambda I)\)  is the zero function on \(\mathbb {T}^d \) .
Moreover, \(\lambda \)  lies in the spectrum \(sp(\Delta )\)  iff
\(det(\hat{\Delta }-\lambda I)\)  has a zero.
For more details see [1]}.
Moreover, we have:
